Using the combined gas law below, identify the variables that would be in the numerator $(A)$ and denominator $(B)$ if you were to rearrange the gas law to solve for final pressure.
[tex]\frac{P_1 V_1}{T_1}=\frac{P_2 V_2}{T_2}[/tex]
[tex]P_2=\frac{A}{B}[/tex]
Numerator $(A)$:
[tex]P_1 V_1 T_2[/tex]
Denominator $(B)$:
[tex]T_1 V_2[/tex]
Answer (2)
The numerator ( A ) when solving for final pressure P 2 is P 1 V 1 T 2 , and the denominator ( B ) is T 1 V 2 . This reflects the relationship as described by the combined gas law. Therefore, we can express P 2 as P 2 = T 1 V 2 P 1 V 1 T 2 .
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Start with the combined gas law equation: T 1 P 1 V 1 = T 2 P 2 V 2 .
Multiply both sides by T 2 : T 1 P 1 V 1 T 2 = P 2 V 2 .
Divide both sides by V 2 : P 2 = T 1 V 2 P 1 V 1 T 2 .
Identify the numerator as P 1 V 1 T 2 and the denominator as T 1 V 2 , so P 2 = T 1 V 2 P 1 V 1 T 2 .
Explanation
Understanding the Combined Gas Law We are given the combined gas law equation:
The Equation T 1 P 1 V 1 = T 2 P 2 V 2
Objective Our goal is to rearrange this equation to solve for P 2 , expressing it in the form P 2 = B A , where A is the numerator and B is the denominator.
Isolating P 2 To isolate P 2 , we need to get rid of V 2 and T 2 on the right side of the equation. We can do this by multiplying both sides by T 2 and dividing both sides by V 2 .
Multiply by T 2 Multiplying both sides by T 2 gives us:
Result T 1 P 1 V 1 T 2 = P 2 V 2
Divide by V 2 Now, divide both sides by V 2 :
Final Equation P 2 = T 1 V 2 P 1 V 1 T 2
Identifying A and B From this, we can identify the numerator A and the denominator B :
Numerator Numerator ( A ) = P 1 V 1 T 2
Denominator Denominator ( B ) = T 1 V 2
Final Answer Therefore, the final answer is:
Result P 2 = T 1 V 2 P 1 V 1 T 2
Examples
The combined gas law is useful in scenarios where you need to predict how the pressure of a gas will change when you alter its volume and temperature, such as in weather forecasting or in industrial processes involving gases. For example, if you have a container of gas at a certain pressure, volume, and temperature, and you want to know what the new pressure will be if you change the volume and temperature, you can use the combined gas law to calculate it. Let's say you have a balloon with an initial pressure P 1 , volume V 1 , and temperature T 1 . If you heat the balloon to a new temperature T 2 and allow it to expand to a new volume V 2 , you can use the combined gas law to find the new pressure P 2 inside the balloon. This is crucial in understanding and predicting the behavior of gases in various real-world applications.
